Method and apparatus for calculating insertion indices for a modular multilevel converter

ABSTRACT

A method for calculating insertion indices for a phase leg of a DC to AC modular multilevel converter. Each phase leg includes two serially connected arms, wherein each arm includes a number of submodules, wherein each submodule can be in a bypass state or a voltage insert mode. The insertion index includes data representing the portion of available submodules that should be in the voltage insert mode. The method includes the steps of: calculating a desired arm voltage for an upper arm connected to the upper DC source common bar and a lower arm connected to the lower DC source common bar, obtaining values representing actual total arm voltages in the upper arm and lower arm, respectively, and calculating modulation indices for the upper and lower arm, respectively, using the respective desired arm voltage and the respective value representing the total actual arm voltage. A corresponding apparatus is also presented.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of pending Internationalpatent application PCT/EP2010/062923 filed on Sep. 3, 2010 whichdesignates the United States and claims the benefit under 35 U.S.C. §119(e) of the U.S. Provisional Patent Application Ser. No. 61/239,859 filedon Sep. 4, 2009. The content of all prior applications is incorporatedherein by reference.

FIELD OF THE INVENTION

The invention relates to the calculation of insertion indices comprisingdata representing the portion of available submodules of a modularmultilevel converter that should be in voltage insert mode.

BACKGROUND OF THE INVENTION

The concept Modular Multilevel Converter (M2C) denotes a class ofVoltage Source Converter (VSC). It has one or several phase legsconnected in parallel between two DC bars, a positive DC+ and a negativeDC−. Each phase leg consists of two series-connected converter arms. Theconnection point between the converter arms constitutes an AC terminalfor the leg.

Each arm consists of a number (N) of submodules. Each submodule has twoterminals. Using these terminals the submodules in each arm areseries-connected so that they form a string. The end terminals of thestring constitute the connection terminals of the arm. By controllingindividual modules in each arm, a voltage corresponding to theaccumulation of insertion voltages can be provided on the AC terminal.

Such a converter is known from DE10103031. In this document, a method toequalize the voltages in the submodule capacitors within the arm isbriefly described. For each arm, a modulator determines when the numberof inserted submodules shall change. The principle for equalizing isthat, at each instant when a change of the number of inserted submodulesis commanded, a selection mechanism chooses the submodule to be insertedor bypassed depending on the actual current direction in the arm(charging or discharging) and the corresponding available submodules inthe arm (bypassed highest voltage/bypassed lowest voltage/insertedhighest voltage/inserted lowest voltage). Such a selection mechanismaims to achieve that the DC voltage across the DC capacitors in thesubmodules are equal, u_(C,SM)(t).

A problem with the prior art is the presence of a circulation currentgoing through the legs between the DC terminals.

SUMMARY OF THE INVENTION

An object of the invention is to reduce circulation currents goingthrough the legs between the DC terminals.

A first aspect is a method for calculating insertion indices for a phaseleg of a DC to AC modular multilevel converter, the converter comprisingone phase leg between upper and lower DC source common bars for eachphase, each phase leg comprising two serially connected arms, wherein anAC output for each phase leg is connected between its two seriallyconnected arms. Each arm comprises a number of submodules, wherein eachsubmodule can be in a bypass state or a voltage insert mode, theinsertion index comprising data representing the portion of availablesubmodules that should be in the voltage insert mode for a particulararm. The method comprises the steps of: calculating a desired armvoltage for an upper arm connected to the upper DC source common bar anda lower arm connected to the lower DC source common bar, obtainingvalues representing actual total arm voltages in the upper arm and lowerarm, respectively, and calculating insertion indices for the upper andlower arm, respectively, using the respective desired arm voltage andthe respective value representing the total actual arm voltage.

The step of calculating desired arm voltages for a phase leg maycomprise calculatingu _(CU)(t)=u _(D)/2−e _(V)(t)−u _(diff)(t)for the upper arm, and calculatingu _(CL)(t)=u _(D)/2+e _(V)(t)−u _(diff)(t)for the lower arm, where u_(CL)(t) represents upper arm voltage, u_(D)represents a voltage between the upper and lower DC source common bars,e_(V)(t) represents a reference inner AC output voltage and u_(diff)(t)represents a control voltage to control a current passing through thewhole phase leg.

The step of calculating a desired arm voltage may comprise calculatingu _(diff)(t)=u _(diff1)(t)+u _(diff2)(t)where u_(diff1)(t) represents a voltage obtained by summing energy inthe arms of the leg and u_(diff2)(t) represents a voltage obtained bycalculating a difference in energy between the arms of the leg.

The step of obtaining a value representing actual arm voltage maycomprise calculatingu _(diff2)(t)=û _(diff2) cos(ω₁ t+ψ)where û_(diff2) represents an error between total upper arm energy andtotal lower arm energy, ω₁ represents the angular velocity of thenetwork frequency and ψ represents the angle given by ψ=∠(R+jω₁L) whereR represents the resistance of the converter arm and L represents theinductance of the converter arm.

The step of obtaining values representing actual arm voltages maycomprise: calculating u_(CU) ^(Σ(t)), actual voltage for the upper arm,using C_(arm), capacitance for the arm, î_(diff0), DC current passingthrough the two serially connected arms of the phase leg, W_(CU)^(Σ)(t), desired average energy in the upper arm, ê_(V), amplitude ofreference for the inner AC output voltage, î_(V), amplitude of AC outputcurrent, φ, a phase difference between i_(V)(t) and e_(V)(t), andcalculating u_(CL) ^(Σ)(t), actual voltage for the lower arm, usingC_(arm), capacitance for the arm, î_(diff0), DC current passing throughthe two serially connected arms of the phase leg, W_(CL) ^(Σ)(t),desired average energy in the lower arm, ê_(V), amplitude of referencefor inner AC output voltage, î_(V), amplitude of AC output current, φ, aphase difference between i_(V)(t) and e_(V)(t).

The step of obtaining a value representing actual arm voltage maycomprise calculating

${\hat{i}}_{{diff}\; 0} = \frac{{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4R{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}}}$where φ represents a phase difference between i_(V)(t) and e_(V)(t),u_(D) represents a voltage between the upper and lower DC source commonbars and R represents the resistance of the converter arm.

The step of obtaining a value representing actual arm voltage maycomprise calculating

${u_{CU}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CU}^{\Sigma}(t)}}{C_{arm}}}$where W_(CU) ^(Σ)(t) represents instantaneous energy in the upper armand is calculated as follows:

${W_{CU}^{\Sigma}(t)} = {W_{{CU}\; 0}^{\Sigma} - {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$and calculating

${u_{CL}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CL}^{\Sigma}(t)}}{C_{arm}}}$where W_(CL) ^(Σ)(t) represents instantaneous energy in the lower armand is calculated as follows:

${W_{CL}^{\Sigma}(t)} = {W_{{CL}\; 0}^{\Sigma} + {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$where ω₁ represents the angular velocity of the network frequency, u_(D)represents a voltage between the upper and lower DC source common bars,R represents the resistance of the converter arm.

The step of obtaining a value representing actual arm voltage maycomprise measuring voltages of the submodules of the arm and summingthese measured voltages.

The insertion index may comprise data representing a direction of theinserted voltage.

A second aspect is an apparatus for calculating insertion indices for aphase leg of a DC to AC modular multilevel converter, the convertercomprising one phase leg between upper and lower DC source common barsfor each phase, each phase leg comprising two serially connected arms,wherein an AC output for each phase leg is connected between its twoserially connected arms, wherein each arm comprises a number ofsubmodules. Each submodule can be in a bypass state or a voltage insertmode, the insertion index comprising data representing the portion ofavailable submodules that should be in the voltage insert mode for aparticular arm. The apparatus comprises a controller arranged tocalculate a desired arm voltage for an upper arm connected to the upperDC source common bar and a lower arm connected to the lower DC sourcecommon bar, to obtain values representing actual total arm voltages inthe upper arm and lower arm, respectively, and to calculate modulationindices for the upper and lower arm, respectively, using the respectivedesired arm voltage and the respective value representing the totalactual arm voltage.

Generally, all terms used in the application are to be interpretedaccording to their ordinary meaning in the technical field, unlessexplicitly defined otherwise herein. All references to “a/an/theelement, apparatus, component, means, step, etc.” are to be interpretedopenly as referring to at least one instance of the element, apparatus,component, means, step, etc., unless explicitly stated otherwise. Thesteps of any method disclosed herein do not have to be performed in theexact order disclosed, unless explicitly stated.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is now described, by way of example, with reference to theaccompanying drawings, in which:

FIG. 1 is a schematic diagram of phase legs and arms;

FIG. 2 is a schematic circuit model of a phase leg of FIG. 1;

FIG. 3 is a Nichols plot for an open loop transfer function;

FIG. 4 is a Nichols plot for an open loop transfer function when a PIDcontroller is used;

FIG. 5 is a Nichols plot for an open loop transfer function when a PIDcontroller, time delay and notch filter is used;

FIG. 6 is a graph showing a simulation result at a step in the referencefor the total energy in the converter leg;

FIG. 7 is a graph showing the simulation result when the current changesfrom 0.1 pu to 1.0 pu in the converter leg; and

FIG. 8 shows a Nichols plot for a balance controller according toanother embodiment.

FIG. 9 is a schematic diagram of an apparatus for calculating insertionindices for a phase leg of a DC to AC modular multilevel converter.

DETAILED DESCRIPTION OF THE INVENTION

The invention will now be described more fully hereinafter withreference to the accompanying drawings, in which certain embodiments ofthe invention are shown. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein; rather, these embodiments are provided byway of example so that this disclosure will be thorough and complete,and will fully convey the scope of the invention to those skilled in theart. Like numbers refer to like elements throughout the description.

In the description in the following, continuous variables are used,corresponding to the simplifying assumption that the arms have infinitenumber of submodules that are switched with infinite switchingfrequency.

FIG. 1 shows an M2C (Modular Multilevel Converter) having a phase leg 7that comprising an upper arm 5 and a lower arm 6. Each arm 5, 6comprising a number of serially connected submodules 9. Each submodule 9comprises a switchable capacitor. An AC output 8 is connected betweenthe upper and lower arms 5, 6. Although only one phase leg 7 is shownhere, the M2C comprises one phase leg 7 for each phase, i.e. three phaselegs 7 for a three phase system, where each phase leg 7 comprises upperand lower arms 5, 6 comprising submodules 9.

An upper DC source common bar (in this case DC+) and a lower DC sourcecommon bar (in this case DC−) for each phase is provided. It is to benoted that the upper and lower DC source common bars can switchpolarity.

Ideally the capacitors keep a constant DC voltage and the AC terminalvoltage is controlled by varying the number of inserted modules in theupper and lower arms. If the voltage between the DC bars is constantthis obviously requires that, in average, the total number of insertedmodules in the two arms remain constant. The arm inductors however willlimit the rate of change of the arm currents, making it possible toaccept minor short deviations from this condition.

Now once the context is presented, define the insertion index, n_(x)(t),for the arm x to be the ratio between the inserted number of submodulesand the total number of available submodules in the arm. The arm voltagethen becomesu _(Cx)(t)=n _(x)(t)u _(C,SM)(t)  (1)

In a simple approach the number of inserted modules in each arm can begenerated by the modulator much in the same way as in PWM modulation forconventional VSCs. Then, in order to generate an inner AC voltage withamplitude ê_(V) the insertion indices for the upper and the lower armsbecome

$\begin{matrix}{{{u_{CU}(t)} = {{{n_{U}(t)}u_{D}\mspace{14mu}{u_{CL}(t)}} = {{n_{L}(t)}u_{D}}}}{{n_{U}(t)} = {{\frac{1 - {\hat{m}\;\cos\;\omega\; t}}{2}\mspace{20mu}{n_{L}(t)}} = \frac{1 - {\hat{m}\;\cos\;\omega\; t}}{2}}}{\hat{m} = \frac{{\hat{e}}_{V}}{\frac{u_{D}}{2}}}} & (2)\end{matrix}$

When the simple modulation approached described above is used and theconverter is loaded on its AC side the desired waveform will bedistorted due to the ripple in the capacitor voltages that will becreated when the load current passes through the converter arms.Specifically a strong second harmonic current will circulate through theconverter leg and the DC side and/or the neighbor phases. This undesiredsecond harmonic current increases the peak of the arms currents andcauses extra losses in the converter arms.

The problem can be solved by generating the insertion indices for thearms, n_(U) and n_(L), in other ways. Such methods would aim to:

-   -   eliminate and/or control the harmonic current in the converter        arms    -   for each arm control the total energy stored in all capacitors        in that arm which is equivalent to control the total voltage of        all capacitors in the arm    -   thereby control the total energy stored in the phase leg as well        as the balance between the upper and the lower arms in the phase        leg

According to the invention the insertion indices n_(U)(t) and n_(L)(t)for the converter arms are being derived in real-time according to thefollowing procedure

-   -   the reference for the converter inner voltage relative the        midpoint of the DC link is given in the form e_(V)(t)=ê_(V) cos        ω₁t; this reference typically is delivered by an AC side        controller operating on AC quantities like output voltage,        current or flux; the converter circuit parameters like arm        resistance and inductance may be used by the controller    -   the desired arm voltages u_(CU)(t) and u_(CL)(t) are calculated        as

$\begin{matrix}{{u_{CU}(t)} = {{\frac{u_{D}}{2} - {e_{V}(t)} - {{u_{diff}(t)}\mspace{14mu}{u_{CL}(t)}}} = {\frac{u_{D}}{2} + {e_{V}(t)} - {u_{diff}(t)}}}} & (3)\end{matrix}$

-   -   where u_(D) is the voltage between the DC rails and u_(diff)(t)        is a control voltage that is created by the control system that        will be described later in the this memo    -   the total capacitor voltages, u_(CU) ^(Σ)(t) and u_(CL) ^(Σ)(t),        of all capacitors in the upper and lower arms respectively, are        measured or estimated as will be described later in this memo    -   the insertion indices are calculated as

$\begin{matrix}{{n_{U}(t)} = {{\frac{u_{CU}(t)}{u_{CU}^{\Sigma}(t)}\mspace{14mu}{n_{L}(t)}} = \frac{u_{CL}(t)}{u_{CL}^{\Sigma}(t)}}} & (4)\end{matrix}$

According to the invention there are two different ways to create thevariables u_(diff)(t), u_(CU) ^(Σ)(t) and u_(CL) ^(Σ)(t).

In this approach the sum of the capacitor voltages in each arm, u_(CU)^(Σ)(t) and u_(CL) ^(Σ)(t), are measured using sensors in thesubmodules. If the voltage sharing between the modules is assumed to beeven the total energies in each arm can be calculated as

$\begin{matrix}{{W_{CU}^{\Sigma}(t)} = {{\frac{C_{arm}}{2}\mspace{14mu}{u_{CU}(t)}^{2}\mspace{14mu}{W_{CL}^{\Sigma}(t)}} = {\frac{C_{arm}}{2}\mspace{14mu}{u_{CL}(t)}^{2}}}} & (5)\end{matrix}$where C_(arm)=C_(submod)/N. Alternatively the energy in each armcapacitor can be calculated individually and the total energy for eacharm then can be created by summing the energies in all submodules ineach arm. The voltage reference component u_(diff)(t) is created as thesum of the output signals from two independent controllersu_(diff)(t)=u_(diff1)(t)+u_(diff2)(t).

The first controller has a reference for the total energy in both armsof the phase leg. The response signal is the measured total energyW_(CU) ^(Σ)(t)+W_(CL) ^(Σ)(t) which may be filtered using e.g. a notchfilter tuned to the frequency 2ω₁ (ω₁ is the network frequency) or anyother filter suppressing the same frequency. The error, i.e. thedifference between the reference and the response signals, is connectedto a controller (normally of type PID) that has the output signalu_(diff1)(t).

The second controller has a reference for the difference between theenergies in the arms in the phase leg. This reference typically is zero,meaning that the energy in the arms in the phase leg shall be balanced.The response signal is created as the measured W_(CU) ^(Σ)(t)−W_(CL)^(Σ)(t), filtered by a notch filter tuned to ω₁ or any other filtersuppressing the same frequency. The error is brought to a controller(typically of P type), which has an output signal û_(diff2). Thecontribution to the total voltage reference u_(diff)(t) is obtained bymultiplying û_(diff2) by a sinusoidal time function cos(ω₁t+ψ), which isphase-shifted relative the inner voltage reference by the angle ψ givenby ψ=∠(R+jω₁L), where R and L are the resistance and inductancerespectively in the converter arm. Thus:u _(diff)(t)=u _(diff1)(t)+û _(diff2) cos(ω₁ t+ψ)  (6)

The first approach to stabilisation of the converter according to theprocedure described in this section is described in more detail inAppendix 1.

Remark 1: The reference for the AC side inner voltage may comprise aminor third harmonic voltage component, which is used to increase theavailable output voltage level in a 3-phase converter. This does notimpact significantly on the behaviour described.

Remark 2: A third reference component may be added to the controlvoltage u_(diff)(t). This component has the purpose of intentionallycreating a second harmonic current in the arms in order to increase theavailable output voltage for loads with certain power factors.

Second Approach, Open-Loop Control

In this approach the AC side current i_(V(t)) is measured. Itsfundamental frequency component is extracted with amplitude and phaserelative the reference inner voltage e_(V(t)) for the converter. Thusthe AC side current can be written asi _(V)(t)=î _(V) cos(ω₁ t+φ)  (7)

Assuming that the converter shall operate ideally in steady-state, i.e.it shall produce undisturbed AC output voltage and the upper and lowerarms shall carry half the AC output current each, it is possible tocalculate the ideal derivative of the energies in each arm. The resultis

$\begin{matrix}{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {{{- {\hat{e}}_{V}}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & (8) \\{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {{{+ {\hat{e}}_{V}}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & (9)\end{matrix}$where î_(diff0) is a DC current circulating through the twoseries-connected arms and the DC supply

$\begin{matrix}{{\hat{i}}_{{diff}\; 0} = \frac{{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4R{\hat{\; e}}_{V}{\hat{i}}_{V}\cos\;\varphi}}}} & (10)\end{matrix}$

When there is only a DC circulating current î_(diff0) then also thecontrol voltage u_(diff)(t) becomes a DC voltage with the valueu_(diff)(t)=Rî_(diff0) so that (3) becomes

$\begin{matrix}{{u_{CU}(t)} = {{\frac{u_{D}}{2} - {e_{V}(t)} - {R\;{\hat{i}}_{{{diff}\; 0}\mspace{14mu}}{u_{CL}(t)}}} = {\frac{u_{D}}{2} + {e_{V}(t)} - {R\;{\hat{i}}_{{diff}\; 0}}}}} & (11)\end{matrix}$

Moreover, equations (8) and (9) can be integrated, each with a freelyselected integration constant, so that

$\begin{matrix}{{W_{CU}^{\Sigma}(t)} = {W_{{CU}\; 0}^{\Sigma} - {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}} & (12) \\{{W_{CL}^{\Sigma}(t)} = {W_{{CL}\; 0}^{\Sigma} + {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}} & (13)\end{matrix}$

Thus the instantaneous energies in each arm can be calculated inreal-time knowing only the references for the inner converter voltageand the actual AC current. The integration constants are the referencesfor the desired average energy in each arm in the phase leg.

But if the energies are known then also the total capacitor voltage inthe arms are know due to the connection equations

$\begin{matrix}{{u_{CU}^{\Sigma}(t)} = {{\sqrt{\frac{2{W_{CU}^{\Sigma}(t)}}{C_{arm}}}\mspace{14mu}{u_{CL}^{\Sigma}(t)}} = \sqrt{\frac{2{W_{CL}^{\Sigma}(t)}}{C_{arm}}}}} & (14)\end{matrix}$

Now the insertion indices valid for the desired steady-state operationcan be calculated using equation (4). Given these insertion indices theenergies in the upper and lower arms converges to the reference valuesgiven as free integration constants in (12) and (13). Normally thesevalues are selected equal for both arms so that balanced operation isobtained. The value of the energy reference is selected to give thedesired total capacitor voltage in each converter arm.

The second approach to stabilisation of the converter according to theprocedure presented in this section is described in more detail inAppendix 2.

Remark 1: If a third harmonic voltage component (to increase theavailable voltage in a 3-phase converter) is added the formulas for theenergies in the upper and lower arm will change somewhat. However theprinciple described in this paper still can be applied.

Remark 2: If even order harmonics are intentionally added to thecirculating current the formulas for the energies in the upper and lowerarms will change somewhat. The principle described above however stillapplies.

-   -   to derive a control strategy that provides main circuit        stability    -   to produce as high AC output voltage as possible with very low        harmonic distortion    -   to control the DC voltages of the capacitors in the modules

Continuous Model

It is of course possible to investigate the M2C converter by simulation.This approach however seems to be quite cumbersome in the sense that itinvolves detailed models of the arms (with tens of semiconductor devicesin each). A lot of data will be generated making it more difficult toextract useful results.

Another approach, which will be followed here, makes use of a modulationprinciple that has been proposed by in DE10103031, in which a selectionmechanism is used to determine which individual module that shall beinserted or bypassed when the number of devices in an arm shall bechanged. The selection is made in dependence of the direction of the armcurrent (or phase current) and a comparison of the DC voltages in thecongregation of modules in each arm, from which the modules having thehighest and lowest voltages are identified.

Simulation has shown that this mechanism successfully keeps the DCvoltages of the module capacitors quite close to each other, even forlow number of modules in each arm (say e.g. five per arm). Thisfunctionality seems to remain even if the total switching frequency islow (a few hundred switchings per second for each semiconductor device).

Now it is assumed that this mechanism is in use and that accordinglythere is no need to look at the DC voltages in the individual modulesany more. The modulation process then can be described in terms of thetotal collective energy in each arm. As the total switching frequency(for all modules in each arm) becomes quite high continuous modellingcan be used. The continuous model is a lot simpler to grasp than thedetailed model and it is an ideal base for understanding the principlesfor the function of the M2C converter and to formulate control laws fordifferent control aspects.

Due to the assumptions static relations exist between the totalcapacitor energy in the upper and lower arm, W_(CU) ^(Σ) and W_(CL)^(Σ), and the corresponding total voltage of all capacitor modules inthe arm, u_(CU) ^(Σ) and u_(CL) ^(Σ). Namely, if it is assumed that theenergy is evenly shared between the modules, this relation becomes

$\begin{matrix}{{W_{CU}^{\Sigma} = {{N\left( {\frac{C}{2}\left( \frac{u_{CU}^{\Sigma}}{N} \right)^{2}} \right)} = {{\frac{C}{2N}u_{CU}^{\Sigma^{2}}\mspace{25mu} u_{CU}^{\Sigma}} = \sqrt{\frac{2N}{C}W_{CU}^{\Sigma}}}}}{W_{CL}^{\Sigma} = {{N\left( {\frac{C}{2}\left( \frac{u_{CL}^{\Sigma}}{N} \right)^{2}} \right)} = {{\frac{C}{2N}u_{CL}^{\Sigma^{2}}\mspace{25mu} u_{CL}^{\Sigma}} = \sqrt{\frac{2N}{C}W_{CL}^{\Sigma}}}}}} & \left( {A\; 1} \right)\end{matrix}$where N is the number of modules per arm and C is the capacitance permodule. In the following we will use the quantity ‘arm capacitance’C_(arm) defined as follows

$\begin{matrix}{C_{arm} = \frac{C}{N}} & \left( {A\; 2} \right)\end{matrix}$Then

$\begin{matrix}{\mspace{14mu}\begin{matrix}{W_{CU}^{\Sigma} = {\frac{C_{arm}}{2}\left( u_{CU}^{\Sigma} \right)^{2}}} & \; & {u_{CU}^{\Sigma} = \sqrt{\frac{2W_{CU}^{\Sigma}}{C_{arm}}}} \\{W_{CL}^{\Sigma} = {\frac{C_{arm}}{2}\left( u_{CL}^{\Sigma} \right)^{2}}} & \Leftrightarrow & {u_{CL}^{\Sigma} = \sqrt{\frac{2W_{CL}^{\Sigma}}{C_{arm}}}}\end{matrix}} & ({A3})\end{matrix}$

Derivation of the Continuous Model

The electrical circuit representing the phase leg of the M2C converteris depicted in FIG. 2. The inserted capacitor voltages, U_(CU) andU_(CL), are created from the total capacitor voltages, u_(CU) ^(Σ) andu_(CL) ^(Σ), respectively, by applying the insertion indices, n_(U) andn_(L), which are controlled by the control system.

$\begin{matrix}{{n_{U} = {{\frac{u_{CU}}{u_{CU}^{\Sigma}}\mspace{14mu} 0} \leq n_{U} \leq 1}}{n_{L} = {{\frac{u_{CL}}{u_{CL}^{\Sigma}}\mspace{14mu} 0} \leq n_{L} \leq 1}}} & \left( {A\; 4} \right)\end{matrix}$

In the following, however, the main circuit model will be formulatedusing the real voltages as variables. If the total capacitor voltages,u_(CU) ^(Σ) and u_(CL) ^(Σ), are measured, the corresponding insertionindices can always be obtained from (A4).

The capacitor modules serve as controlled electromotive forces in thecircuit. Let the total energy in the capacitors be W_(CU) ^(Σ) andW_(CL) ^(Σ) in the upper and lower arms respectively. Inspection of thecircuit model in FIG. 2 immediately yields

$\begin{matrix}{{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {i_{U}u_{CU}}}{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {{- i_{L}}u_{CL}}}} & \left( {A\; 5} \right)\end{matrix}$

In order to gain some more insight it is helpful separate the armscurrents in two parts. One part emerges from the AC current, whichnaturally separates into two halves, one passing though the upper andone passing through the lower arm. The deviation from this “ideal”condition is described by a difference current i_(diff) which passesthrough the series-connected arms and the DC source.

Define

$\begin{matrix}{{i_{U} = {\frac{i_{V}}{2} + i_{diff}}}{i_{L} = {\left. {\frac{i_{V}}{2} - i_{diff}}\Leftrightarrow i_{diff} \right. = \frac{i_{U} - i_{L}}{2}}}} & \left( {A\; 6} \right)\end{matrix}$

The circuit in FIG. 2 now gives the equations

$\begin{matrix}{{\frac{u_{D}}{2} - {R\left( {\frac{i_{V}}{2} + i_{diff}} \right)} - {L\left( {{\frac{1}{2}\frac{\mathbb{d}i_{V}}{\mathbb{d}t}} + \frac{\mathbb{d}i_{diff}}{\mathbb{d}t}} \right)} - u_{CU} - u_{V}} = {{0 - \frac{u_{D}}{2} - {R\left( {\frac{i_{V}}{2} - i_{diff}} \right)} - {L\left( {{\frac{1}{2}\frac{\mathbb{d}i_{V}}{\mathbb{d}t}} - \frac{\mathbb{d}i_{diff}}{\mathbb{d}t}} \right)} + u_{CL} - u_{V}} = 0}} & \left( {A\; 7} \right)\end{matrix}$

Adding and subtracting the equations give the results

$\begin{matrix}{{u_{V} = {\frac{u_{CL} - u_{CU}}{2} - {\frac{R}{2}i_{V}} - {\frac{L}{2}\frac{\mathbb{d}i_{V}}{\mathbb{d}t}}}}{{{L\frac{\mathbb{d}i_{diff}}{\mathbb{d}t}} + {Ri}_{diff}} = {\frac{u_{D}}{2} - \frac{u_{CL} + u_{CU}}{2}}}} & \left( {A\; 8} \right)\end{matrix}$

These equations show that

-   -   the AC voltage only depends on the AC current i_(V) and the        difference between the arm voltages u_(CL) and u_(CU)    -   the arm voltage difference acts as an inner AC voltage in the        converter and the inductance L and resistance R form a fix,        passive inner impedance for the AC current    -   the difference current i_(diff) only depends on the DC link        voltage and the sum of the arm voltages    -   the difference current i_(diff) can be controlled independently        of the AC side quantities by subtracting the same voltage        contributions to both arms

Define

$\begin{matrix}{\mspace{31mu}\begin{matrix}{e_{V} = \frac{u_{CL} - u_{CU}}{2}} & \; & {u_{CU} = {\frac{u_{D}}{2} - e_{V} - u_{diff}}} \\{u_{diff} = \frac{u_{D} - u_{CL} - u_{CU}}{2}} & \Leftrightarrow & {u_{CL} = {\frac{u_{D}}{2} + e_{V} - u_{diff}}}\end{matrix}} & \left( {A\; 9} \right)\end{matrix}$where e_(V) is the desired inner voltage in the AC voltage source andu_(diff) is a voltage that controls the difference current i_(diff).

Then (8) becomes

$\begin{matrix}{{u_{V} = {e_{V} - {\frac{R}{2}i_{V}} - {\frac{L}{2}\frac{\mathbb{d}i_{V}}{\mathbb{d}t}}}}{{{L\frac{\mathbb{d}i_{diff}}{\mathbb{d}t}} + {Ri}_{diff}} = u_{diff}}} & \left( {A\; 10} \right)\end{matrix}$

Inserting equations (A6) and (A9) in (A5) yields

$\begin{matrix}{{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {\left( {\frac{i_{V}}{2} + i_{diff}} \right)\left( {\frac{u_{D}}{2} - e_{V} - u_{diff}} \right)}}{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {\left( {{- \frac{i_{V}}{2}} + i_{diff}} \right)\left( {\frac{u_{D}}{2} + e_{V} - u_{diff}} \right)}}} & \left( {A\; 11} \right)\end{matrix}$

It makes sense to investigate the total energy stored in all capacitorbanks in the whole leg and to examine the balance between the energy inthe upper and the lower arm.

Define

$\begin{matrix}{{W_{C}^{\Sigma} = {{W_{CU}^{\Sigma} + {W_{CL}^{\Sigma}\mspace{45mu} W_{CU}^{\Sigma}}} = \frac{W_{C}^{\Sigma} + W_{C}^{\Delta}}{2}}}{W_{C}^{\Delta} = {\left. {W_{CU}^{\Sigma} - W_{CL}^{\Sigma}}\Leftrightarrow W_{CL}^{\Sigma} \right. = \frac{W_{C}^{\Sigma} - W_{C}^{\Delta}}{2}}}} & \left( {A\; 12} \right)\end{matrix}$

The result is

$\begin{matrix}{{\frac{\mathbb{d}W_{C}^{\Sigma}}{\mathbb{d}t} = {{\left( {u_{D} - {2\; u_{diff}}} \right)i_{diff}} - {e_{V}i_{V}}}}{\frac{\mathbb{d}W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}\; e_{V}i_{diff}} + {\left( {\frac{u_{D}}{2} - u_{diff}} \right)i_{V}}}}} & \left( {A\; 13} \right)\end{matrix}$

Equation (A13) indicates that the total energy in both arms as well asthe energy balance between the upper and lower arms can be controlledprimarily by i_(diff), which is in its turn controlled by u_(diff)through (A10).

The term in the upper equation in (A13) is recognized as theinstantaneous power delivered to the AC side.p _(V) =e _(V) i _(V)  (A14)

Steady State Solution

We shall look at the special case where the AC emf and current is given.Thus let

$\begin{matrix}{{e_{V} = {{\hat{e}}_{V}\cos\mspace{11mu}\omega\; t}}{i_{V} = {{\hat{i}}_{V}\mspace{11mu}{\cos\left( {{\omega\; t} + \varphi} \right)}}}} & ({A15})\end{matrix}$

Assume that there is a solution where the difference current i_(diff) isa pure dc component. Thusi _(diff)(t)=î _(diff)  (A16)

Then, according to (A10)u _(diff)(t)=Rî _(diff)  (A17)

The derivative of the total and difference energies the according to(A13) become

$\begin{matrix}{{\frac{\mathbb{d}W_{C}^{\Sigma}}{\mathbb{d}t} = {{\left( {u_{D} - {2\; R{\hat{i}}_{diff}}} \right){\hat{i}}_{diff}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{2}\left\lbrack {{\cos\;\varphi} + {\cos\left( {{2\omega\; t} + \varphi} \right)}} \right\rbrack}}}{\frac{\mathbb{d}W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}\;{\hat{e}}_{V}{\hat{i}}_{diff}\cos\;\omega\; t} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{diff}}} \right){\hat{i}}_{V}{\cos\left( {{\omega\; t} + \varphi} \right)}}}}} & \left( {A\; 18} \right)\end{matrix}$

From (A18) some observations can immediately be made

-   -   the derivative of the total energy contains only a constant and        a component having double network frequency    -   the derivative of the difference energy only contains components        having network frequency

Steady-state condition requires that the constant component of the totalenergy derivative disappears so that

$\begin{matrix}{{{{\left( {u_{D} - {2\; R{\hat{i}}_{diff}}} \right){\hat{i}}_{diff}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{2}\cos\;\varphi}} = 0}{{\hat{i}}_{diff} = \frac{{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4\; R{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}}}}} & \left( {A\; 19} \right)\end{matrix}$

With this difference current the remaining term becomes

$\begin{matrix}{\frac{\mathbb{d}W_{C}^{\Sigma}}{\mathbb{d}t} = {{- \frac{{\hat{e}}_{V}{\hat{i}}_{V}}{2}}{\cos\left( {{2\omega\; t} + \varphi} \right)}}} & \left( {A\; 20} \right)\end{matrix}$

The steady-state average energy can be freely selected so that the totalenergy in steady-state becomes

$\begin{matrix}{{W_{C}^{\Sigma}(t)} = {W_{C\; 0}^{\Sigma} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4\omega}{\sin\left( {{2\omega\; t} + \varphi} \right)}}}} & \left( {A\; 21} \right)\end{matrix}$

The expression for the difference energy In (A18) can be directlyintegrated, also with a freely selectable integration constant (whichnormally shall be zero)

$\begin{matrix}{{W_{C}^{\Delta}(t)} = {W_{C\; 0}^{\Delta} - {\frac{2{\hat{e}}_{V}{\hat{i}}_{diff}}{\omega}\sin\;\omega\; t} + {\frac{\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{diff}}} \right){\hat{i}}_{V}}{\omega}{\sin\left( {{\omega\; t} + \varphi} \right)}}}} & \left( {A\; 22} \right)\end{matrix}$

The investigation shows that

-   -   solutions of the desired type, i.e. with a difference component        having only a DC component, exist with freely selectable energy        levels in each arm    -   the steady state solutions for the energy time functions contain        only a double frequency component in the total energy and a        fundamental frequency component in the difference energy

Linearised Model for Control Studies

Let us go back and linearise the equations (A13) around a steady statepoint as described in the preceding section. Assume that the DC linkvoltage is constant. The

$\begin{matrix}{\mspace{79mu}{{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Sigma}}{\mathbb{d}t} = {{\left( {u_{D} - {2\; R{\hat{i}}_{diff}}} \right)\Delta\; i_{diff}} - {2\;{\hat{i}}_{diff}\Delta\; u_{diff}} - {\Delta\; p_{V}}}}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}e_{V}\Delta\; i_{diff}} - {i_{V}\Delta\; u_{diff}} - {2\;{\hat{i}}_{diff}\Delta\; e_{V}} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{diff}}} \right)\Delta\; i_{V}}}}}} & \left( {A\; 23} \right)\end{matrix}$

Further the differential equation that governs i_(diff) as function ofu_(diff) applies so that

$\begin{matrix}{{{L\frac{{\mathbb{d}\Delta}\; i_{diff}}{\mathbb{d}t}} + {R\;\Delta\; i_{diff}}} = {\Delta\; u_{diff}}} & \left( {A\; 24} \right)\end{matrix}$

Stability Requirements

When the AC side current is stiff (A23) reduces to

$\begin{matrix}{{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Sigma}}{\mathbb{d}t} = {{\left( {u_{D} - {2\; R{\hat{i}}_{diff}}} \right)\Delta\; i_{diff}} - {2\;{\hat{i}}_{diff}\Delta\; u_{diff}}}}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}e_{V}\Delta\; i_{diff}} - {i_{V}\Delta\; u_{diff}}}}} & \left( {A\; 25} \right)\end{matrix}$

The linearized equations (A25) show that any control system, which makesthe sum of the inserted voltages, u_(CL) and u_(CU), perfectly match thevoltage u_(D) on the DC side, i.e. makes u_(diff)≡0, also makes thedifference current become zero causing the derivatives of the energiesin the arms to vanish. The main circuit in the converter then ismarginally stable. Thus is not sufficient to select the insertedvoltages in (A9) according to the desired e_(V), but an u_(diff) thatcreates stability must also be provided.

Control Law for the Total Capacitor Energy

The equation for the total energy equation can be formulated in theLaplace domain

$\begin{matrix}\begin{matrix}{{\Delta\;{W_{C}^{\Sigma}(s)}} = {{{\frac{1}{s}\left\{ {\frac{\left( {u_{D} - {2\; R{\hat{i}}_{diff}}} \right)}{R + {sL}} - {2\;{\hat{i}}_{diff}}} \right\}\Delta\;{u_{diff}(s)}} - \frac{\Delta\;{p_{V}(s)}}{s}} =}} \\{= {{\frac{u_{D} - {4\; R{\hat{i}}_{diff}} - {2\; L{\hat{i}}_{diff}s}}{\left( {R + {sL}} \right)s}\Delta\;{u_{diff}(s)}} - \frac{\Delta\;{p_{V}(s)}}{s}}}\end{matrix} & \left( {A\; 26} \right)\end{matrix}$

Applying a Proportional Gain in an Energy ControllerΔu _(diff)(s)=K _(P) {ΔW _(C) ^(Σref)(s)−ΔW _(C) ^(Σ)(s)}  (A27)yields

$\begin{matrix}{{\Delta\;{W_{C}^{\Sigma}(s)}} = {{\frac{\left\lbrack {u_{D} - {4\; R{\hat{i}}_{diff}} - {2\; L{\hat{i}}_{diff}s}} \right\rbrack K_{p}}{{\left( {R + {sL}} \right)s} + {\left\lbrack {u_{D} - {4\; R{\hat{i}}_{{diff}\;}} - {2\; L{\hat{i}}_{diff}s}} \right\rbrack K_{P}}}\Delta\;{W_{C}^{\Sigma\;{ref}}(s)}} - {\quad{{- \frac{R + {sL}}{{\left( {R + {sL}} \right)s} + {\left\lbrack {u_{D} - {4\; R{\hat{i}}_{diff}} - {2\; L{\hat{i}}_{diff}s}} \right\rbrack K_{P}}}}\Delta\;{p_{V}(s)}}}}} & \left( {A\; 28} \right)\end{matrix}$

The poles in the above transfer functions are mainly determined by

$\begin{matrix}{{{s_{1,2}^{2}L} + {u_{D}K_{p}}} = {\left. 0\Leftrightarrow s_{1,2} \right. = {{\pm j}\sqrt{\frac{u_{D}K_{p}}{L}}}}} & \left( {A\; 29} \right)\end{matrix}$

The control system is investigated for a converter leg in an exampleconverter with the main parameters given in Table 1.

TABLE 1 Example converter main data 3-ph rated power 30 MVA ratedfrequency 50 Hz line-line voltage 13.8 kV rms rated phase current 1255 Arms arm capacitance 500 μF/arm arm inductance 3 mH arm resistance 100 mΩ

FIG. 3 shows the Nichols plot for the open loop transfer function in(A26) with the proportional gain K_(P)=0.001 V/J. The curve is almostindependent of the active load.

As expected is the phase margin at 90 rad/s quite small, which meansthat the response will be quite oscillatory. In FIG. 4 the Nichols plotis shown when a PID controller is used.

The selected transfer function is given by

$\begin{matrix}{{F^{\Sigma}(s)} = {K_{P}\left( {1 + \frac{K_{I}}{s} + \frac{{sT}_{D}}{1 + {sT}_{F}}} \right)}} & \left( {A\; 30} \right)\end{matrix}$

with K_(P)=0.002 V/J, K_(I)=20 s⁻¹, T_(D)=10 ms, T_(F)=2 ms

It has been shown that the total energy response signal contains afrequency component with twice the network frequency. This component canbe removed from the controller response using a notch filter. Further itis advisable to assume that a delay occurs in the measured total energy(total capacitor voltage). FIG. 5 shows the corresponding Nicols'sdiagram where the notch filter and the time delay have been included.

The transfer function in FIG. 5 is

$\begin{matrix}{{F^{\Sigma}(s)} = {{K_{P}\left( {1 + \frac{K_{I}}{s} + \frac{{sT}_{D}}{1 + {sT}_{F}}} \right)}{\mathbb{e}}^{- {sT}_{del}}\frac{s^{2} + \left( {2\omega_{1}} \right)^{2}}{s^{2} + {2{\zeta\left( {2\omega_{1}} \right)}s} + \left( {2\omega_{1}} \right)^{2}}}} & \left( {A\; 31} \right)\end{matrix}$

with K_(P)=0.002 V/J, K_(I)=20 s⁻¹, T_(D)=10 ms, T_(F)=2 ms, T_(del)=1ms, ζ=0.05

FIG. 6 shows the simulation result at a step in the reference for thetotal energy in the converter leg.

Equation (A28) shows that the energy control system having only aproportional feedback will have a static error

$\begin{matrix}{\frac{\Delta\; W_{C}^{\Sigma}}{\Delta\; p_{V}} = \frac{- R}{K_{P}\left( {u_{D} - {4\; R{\hat{i}}_{diff}}} \right)}} & \left( {A\; 32} \right)\end{matrix}$

For the values in Table 1 together with K_(P)=0.002 V/J this energydependence becomes approximately 0.002 J/W. Each leg of the converterhandles about 10 MW causing the energy drop to be about 20 kJ (out ofabout 312 kJ) per leg.

FIG. 7 shows the simulation result when the current changes from 0.1 puto 1.0 pu in the converter leg.

Control Law for Balancing the Capacitor Energies in the Arms

The general differential equation governing the balance between theenergies in the upper and the lower arm was derived in (A13)

$\begin{matrix}{\frac{\mathbb{d}W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}e_{V}i_{diff}} + {\left( {\frac{u_{D}}{2} - u_{diff}} \right)i_{V}}}} & \left( {A\; 33} \right)\end{matrix}$

It was linearised in (A23)

$\begin{matrix}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}e_{V}\Delta\; i_{diff}} - {i_{V}\Delta\; u_{diff}} - {2{\hat{i}}_{diff}\Delta\; e_{V}} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{diff}}} \right)\Delta\; i_{V}}}} & \left( {A\; 34} \right)\end{matrix}$

If we consider linearising around the steady state solution defined by(A15) the linearised equation becomes

$\begin{matrix}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}{\hat{e}}_{V}\cos\;\omega\; t\;\Delta\; i_{diff}} - {{\hat{i}}_{V}{\cos\left( {{\omega\; t} + \varphi} \right)}\Delta\; u_{diff}} - {2{\hat{i}}_{diff}\Delta\; e_{V}} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{diff}}} \right)\Delta\; i_{V}}}} & \left( {A\; 35} \right)\end{matrix}$

Assume first that the AC side quantities are constant. Then

$\begin{matrix}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{{- 2}{\hat{e}}_{V}\cos\;\omega\; t\;\Delta\; i_{diff}} - {{\hat{i}}_{V}{\cos\left( {{\omega\; t} + \varphi} \right)}\Delta\; u_{diff}}}} & \left( {A\; 36} \right)\end{matrix}$

Further assume that the controller produces a fundamental frequencysinusoidal signal with phase ζ relative the inner emf in the converterlegu _(diff)(t)=û _(diff)(t)cos(ωt+ξ)  (A37)

Using the quasi-stationary solution to (A10) yields

$\begin{matrix}{{{{\hat{i}}_{diff}(t)} = {\frac{{\hat{u}}_{diff}}{\sqrt{R^{2} + \left( {\omega\; L} \right)^{2}}}{\cos\left( {{\omega\; t} + \xi - \eta} \right)}}}{\eta = {\arg\left( {R + {{j\omega}\; L}} \right)}}} & \left( {A\; 38} \right)\end{matrix}$

Inserting in (A36) we get

$\begin{matrix}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {\left\{ {{{- \frac{2{\hat{e}}_{V}}{\sqrt{R^{2} + \left( {\omega\; L} \right)^{2}}}}{\cos\left( {\omega\; t} \right)}{\cos\left( {{\omega\; t} + \xi - \eta} \right)}} - {{\hat{i}}_{V}{\cos\left( {{\omega\; t} + \varphi} \right)}{\cos\left( {{\omega\; t} + \xi} \right)}}} \right\}\Delta{\hat{u}}_{diff}}} & ({A39})\end{matrix}$

The products of the cosine functions in (A39) are DC quantities andterms with the double network frequency. These components are

$\begin{matrix}{\mspace{79mu}{{a^{({d\; c})} = {{\frac{- {\hat{e}}_{V}}{\sqrt{R^{2} + \left( {\omega\; L} \right)^{2}}}{\cos\left( {\xi - \eta} \right)}} - {\frac{{\hat{i}}_{V}}{2}{\cos\left( {\xi - \varphi} \right)}}}}{a^{({2\omega})} = {{\frac{- {\hat{e}}_{V}}{\sqrt{R^{2} + \left( {\omega\; L} \right)^{2}}}{\cos\left( {{2\omega\; t} + \xi - \eta} \right)}} - {\frac{{\hat{i}}_{V}}{2}{\cos\left( {{2\;\omega\; t} + \xi + \varphi} \right)}}}}}} & ({A40})\end{matrix}$

The relation between the two terms at various frequencies has been foundto show that the first term dominates completely even for operatingfrequencies down to 5 Hz. Therefore it is sufficient to consider thefirst term. The maximum DC component then is obtained whenξ=η=arg(R+jωL)  (A41)

With this selection of the argument for the inserted difference voltagewe get the simplified formula

$\begin{matrix}{\frac{{\mathbb{d}\Delta}\; W_{C}^{\Delta}}{\mathbb{d}t} = {{- \frac{{\hat{e}}_{V}}{\sqrt{R^{2} + \left( {\omega\; L} \right)^{2}}}}\Delta\;{\hat{u}}_{diff}}} & ({A42})\end{matrix}$

A proportional controller is sufficient to control the balance betweenthe energies as the function is indeed just an integrator. However, themeasured difference energy has a strong fundamental frequency component,which should be eliminated in the response to the regulator. Thus thetransfer function in the balancing controller is given by

$\begin{matrix}{{F^{\Delta}(s)} = {K_{P\;\Delta}\frac{s^{2} + \omega_{1}^{2}}{s^{2} + {2{\zeta\omega}_{1}} + \omega_{1}^{2}}}} & ({A43})\end{matrix}$

FIG. 8 shows the Nichols plot for the balance controller with parametersaccording to Table 1 and with control parameters K_(PΔ)=−0.005 V/J,t_(Del)=1 MS, ζ=0.1.

FIG. 5 shows that the closed loop for the energy controller has unitygain up to about 300 rad/s and that it amplifies frequencies in therange 100-200 rad/s with more than 3 dB. Therefore the gain in thebalance controller has been kept low for these frequencies in order toavoid interaction between the two controllers.

Appendix 2: Description of Open Loop Control System

The aim of the investigation is to describe an M2C system where themodulation operates in open-loop mode. The meaning of the name“open-loop” in this context is that the modulation system does notmeasure the total voltage of the capacitors in the phase leg arms.Rather these total voltages are estimated in run-time using the desiredAC emf and the measured AC current. The reference for the inserted armvoltages are obtained assuming that the instantaneous AC emf and ACcurrent are steady state values. Further it is assumed that a voltagesharing system is provided to distribute the total arm voltage in eacharm evenly between all modules that constitute the arm.

Steady State Analysis

The starting point is that the converter produces a sinusoidal emfe _(V) =ê _(V) cos ω₁ t  (B1)and is loaded with a sinusoidal phase currenti _(V) =î _(V) cos(ω₁ t+φ)  (B2)

Under ideal conditions the arm currents only contains a DC componentî_(diff0) so that the arm currents become

$\begin{matrix}{{i_{U} = {{\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} + {\hat{i}}_{{diff}\; 0}}}{i_{L} = {{\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\hat{i}}_{{diff}\; 0}}}} & ({B3})\end{matrix}$

When the difference current is î_(diff0) the difference voltage becomesu_(diff)=Rî_(diff0) so that the arm voltages become

$\begin{matrix}{{u_{CU} = {\frac{u_{D}}{2} - {{\hat{e}}_{V}\cos\;\omega_{1}t} - {R{\hat{i}}_{{diff}\; 0}}}}{u_{CL} = {\frac{u_{D}}{2} + {{\hat{e}}_{V}\cos\;\omega_{1}t} - {R{\hat{i}}_{{diff}\; 0}}}}} & ({B4})\end{matrix}$

The derivatives of the arm energies are

$\begin{matrix}{{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {u_{CU}i_{U}}}{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {{- u_{CL}}i_{L}}}} & ({B5})\end{matrix}$

Inserting the expressions in (B3) and (B4) yields

$\begin{matrix}{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {{{\left\{ {{\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right){\hat{i}}_{{diff}\; 0}} - \frac{{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}{4}} \right\}--}{\hat{e}}_{V}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & ({B6}) \\{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {\left\{ {{\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right){\hat{i}}_{{diff}\; 0}} - \frac{{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}{4}} \right\} - {{+ {\hat{e}}_{V}}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & ({B7})\end{matrix}$

In steady state the DC term must be zero. This condition allows us todetermine the DC component to

$\begin{matrix}{{\hat{i}}_{{diff}\; 0} = \frac{{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4R{\hat{e}}_{V}{\hat{i}}_{V}\cos\;\varphi}}}} & ({B8})\end{matrix}$

Thus, in steady state, the energy variations are

$\begin{matrix}{\frac{\mathbb{d}W_{CU}^{\Sigma}}{\mathbb{d}t} = {{{- {\hat{e}}_{V}}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & ({B9}) \\{\frac{\mathbb{d}W_{CL}^{\Sigma}}{\mathbb{d}t} = {{{+ {\hat{e}}_{V}}{\hat{i}}_{{diff}\; 0}\cos\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2}{\cos\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4}{\cos\left( {{2\omega_{1}t} + \varphi} \right)}}}} & ({B10})\end{matrix}$

These formulas can immediately be integrated to obtain the instantaneousenergy variations. Note that a freely selectable integration constantappears in each expression. Thus

$\begin{matrix}{{W_{CU}^{\Sigma}(t)} = {W_{{CU}\; 0}^{\Sigma} - {\frac{{\hat{e}}_{V}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\;\omega_{1}t} + \varphi} \right)}}}} & ({B11}) \\{{W_{CL}^{\Sigma}(t)} = {W_{{CL}\; 0}^{\Sigma} + {\frac{{\hat{e}}_{V}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{4\omega_{1}}{\sin\left( {{2\;\omega_{1}t} + \varphi} \right)}}}} & ({B12})\end{matrix}$

The total capacitor voltages now are given by

$\begin{matrix}{{{u_{CU}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CU}^{\Sigma}(t)}}{C_{arm}}}}{{u_{CL}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CL}^{\Sigma}(t)}}{C_{arm}}}}} & ({B13})\end{matrix}$and they can be used to determine the insertion indices in run-timeaccording to

$\begin{matrix}{{{n_{U}(t)} = \frac{u_{CU}(t)}{u_{CU}^{\Sigma}(t)}}{{n_{L}(t)} = \frac{u_{CL}(t)}{u_{CL}^{\Sigma}(t)}}} & ({B14})\end{matrix}$

Open-Loop Control

The idea of the open-loop control mode is to

-   -   measure the AC terminal current    -   extract the amplitude and phase relative the created emf in the        converter as in (B2)    -   perform the calculation as described above    -   utilize the so obtained insertion indices according to (B14) in        the converter

FIG. 9 shows an apparatus 100 for calculating insertion indices for aphase leg of the M2C illustrated in FIG. 1. The apparatus 100 comprisesa controller 102 arranged to calculate a desired arm voltage for theupper arm 5 connected to the upper DC source common bar and the lowerarm 6 connected to the lower DC source common bar, to obtain valuesrepresenting actual total arm voltages in the upper arm 5 and lower arm6, respectively, and to calculate modulation indices for the upper arm 5and lower arm 6, respectively, using the respective desired arm voltageand the respective value representing the total actual arm voltage. Thecontroller 102 performs the steps of calculating the desired armvoltages for a phase leg, obtaining values representing actual armvoltages, and calculating insertion indices by using the methodsdiscussed above.

The invention has mainly been described above with reference to a fewembodiments. However, as is readily appreciated by a person skilled inthe art, other embodiments than the ones disclosed above are equallypossible within the scope of the invention, as defined by the appendedpatent claims.

1. A method for calculating insertion indices for a phase leg of a DC toAC modular multilevel converter via a control apparatus, said controlapparatus having at east one controller, the converter comprising onephase leg between upper and lower DC source common bars for each phase,each phase leg comprising two serially connected arms, wherein an ACoutput for each phase leg is connected between the two seriallyconnected arms, wherein each arm comprises a number of submodules,wherein each submodule can be in a bypass state or a voltage insertmode, the insertion index comprising data representing a portion ofavailable submodules that should be in the voltage insert mode for aparticular arm, the method comprising the steps of: calculating, in saidcontroller, a desired arm voltage for an upper arm connected to theupper DC source common bar and a lower arm connected to the lower DCsource common bar, obtaining, in said controller, values representingactual total arm voltages in the upper arm and lower arm, respectively,and calculating, in said controller, insertion indices for the upper andlower arm, respectively, using the respective desired arm voltage andthe respective value representing the total actual arm voltage; whereinthe step of calculating desired arm voltages for a phase leg comprisescalculatingu _(CU)(t)=u _(D)/2−e _(V)(t)−u _(diff)(t) for the upper arm, andcalculatingu _(CL)(t)=u _(D)/2+e _(V)(t)−u _(diff)(t) for the lower arm, whereu_(CU)(t) represents desired upper arm voltage where u_(CL)(t)represents desired lower arm voltage, u_(D) represents a voltage betweenthe upper and lower DC source common bars, e_(V)(t) represents areference inner AC output voltage and u_(diff)(t) represents a controlvoltage to control a current passing through the whole phase leg, andcalculatingu _(diff)(t)=u _(diff1)(t)+u _(diff2)(t) where u_(diff1)(t) represents avoltage obtained by summing energy in the arms of the leg andu_(diff2)(t) represents a voltage obtained by calculating a differencein energy between the arms of the leg.
 2. The method according to claim1, wherein the step of obtaining a value representing actual arm voltagecomprises calculatingu _(diff2)(t)=û _(diff2) cos(ω₁ t+ψ) where û_(diff2) represents adifference between total upper arm energy and total lower arm energy, ω₁represents the angular velocity of the network frequency and ψrepresents the angle given by ψ=∠(R+jω₁L) where R represents theresistance of the converter arm and L represents the inductance of theconverter arm.
 3. The method according to claim 1, wherein the step ofobtaining a value representing actual total arm voltage comprisesmeasuring voltages of the submodules of the arm and summing thesemeasured voltages.
 4. The method according to claim 1, wherein theinsertion index comprises data representing a direction of the insertedvoltage.
 5. A method for calculating insertion indices for a phase legof a DC to AC modular multilevel converter via a control apparatus, saidcontrol apparatus having at least one controller, the convertercomprising one phase leg between upper and lower DC source common barsfor each phase, each phase leg comprising two serially connected arms,wherein an AC output for each phase leg is connected between the twoserially connected arms, wherein each arm comprises a number ofsubmodules, wherein each submodule can be in a bypass state or a voltageinsert mode, the insertion index comprising data representing a portionof available submodules that should be in the voltage insert mode for aparticular arm, the method comprising the steps of: calculating, in saidcontroller, a desired arm voltage for an upper arm connected to theupper DC source common bar and a lower arm connected to the lower DCsource common bar, obtaining, in said controller, values representingactual total arm voltages in the upper arm and lower arm, respectively,and calculating, in said controller, insertion indices for the upper andlower arm, respectively, using the respective desired arm voltage andthe respective value representing the total actual arm voltage; whereinthe step of calculating desired arm voltages for a phase leg comprisescalculatingu _(CU)(t)=u _(D)/2−e _(V)(t)−u _(diff)(t) for the upper arm, andcalculatingu _(CL)(t)=u _(D)/2+e _(V)(t)−u _(diff)(t) for the lower arm, whereu_(CU)(t) represents desired upper arm voltage where u_(CL)(t)represents desired lower arm voltage, u_(D) represents a voltage betweenthe upper and lower DC source common bars, e_(V)(t) represents areference inner AC output voltage and u_(diff)(t) represents a controlvoltage to control a current, iV(t), passing through the whole phaseleg, wherein the step of obtaining values representing actual armvoltages comprises: calculating u_(CU) ^(Σ)(t), actual total voltage forthe upper arm, using C_(arm), capacitance for the upper arm, î_(diff0),DC current passing through the two serially connected arms of the phaseleg, W_(CU) ^(Σ)(t), desired average energy in the upper arm, ê_(V),amplitude of reference for the inner AC output voltage, î_(V), amplitudeof AC output current, φ, a phase difference between i_(V)(t) ande_(V)(t), and calculating u_(CL) ^(Σ)(t), actual total voltage for thelower arm, using C_(arm), capacitance for the lower arm, î_(diff0), DCcurrent passing through the two serially connected arms of the phaseleg, W_(CL) ^(Σ)(t), desired average energy in the lower arm, ê_(V),amplitude of reference for inner AC output voltage, î_(V), amplitude ofAC output current, φ, a phase difference between i_(V)(t) and e_(V)(t).6. The method according to claim 5, wherein the step of obtaining avalue representing actual arm voltage comprises calculating${\hat{i}}_{{diff}\; 0} = \frac{{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4R{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}}}$where φ represents a phase difference between i_(V)(t) and e_(V)(t),u_(D) represents a voltage between the upper and lower DC source commonbars and R represents the resistance of the converter arm.
 7. The methodaccording to claim 6, wherein the step of obtaining a value representingactual arm voltage comprises calculating${u_{CU}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CU}^{\Sigma}(t)}}{C_{arm}}}$where W_(CU) ^(Σ)(t) is calculated as follows:${W_{CU}^{\Sigma}(t)} = {W_{{CU}\; 0}^{\Sigma} - {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$and calculating${u_{CL}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CL}^{\Sigma}(t)}}{C_{arm}}}$where W_(CL) ^(Σ)(t) is calculated as follows:${W_{CL}^{\Sigma}(t)} = {W_{{CL}\; 0}^{\Sigma} + {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$where ω₁ represents the angular velocity of the network frequency.
 8. Anapparatus for calculating insertion indices for a phase leg of a DC toAC modular multilevel converter, the converter comprising one phase legbetween upper and lower DC source common bars for each phase, each phaseleg comprising two serially connected arms, wherein an AC output foreach phase leg is connected between the two serially connected arms,wherein each arm comprises a number of submodules, wherein eachsubmodule can be in a bypass state or a voltage insert mode, theinsertion index comprising data representing a portion of availablesubmodules that should be in the voltage insert mode for a particulararm, the apparatus comprises: a controller arranged to calculate adesired arm voltage for an upper arm connected to the upper DC sourcecommon bar and a lower arm connected to the lower DC source common bar,to obtain values representing actual total arm voltages in the upper armand lower arm, respectively, and to calculate insertion indices for theupper and lower arm, respectively, using the respective desired armvoltage and the respective value representing the total actual armvoltage; wherein the calculating of desired arm voltages for a phase legcomprises calculatingu _(CU)(t)=u _(D)/2−e _(V)(t)−u _(diff)(t) for the upper arm, andcalculatingu _(CL)(t)=u _(D)/2+e _(V)(t)−u _(diff)(t) for the lower arm, whereu_(CU)(t) represents desired upper arm voltage, u_(CL)(t) representsdesired lower arm voltage, u_(D) represents a voltage between the upperand lower DC source common bars, e_(V)(t) represents a reference innerAC output voltage and u_(diff)(t) represents a control voltage tocontrol a current passing through the whole phase leg, and calculatingu _(diff)(t)=u _(diff1)(t)+u _(diff2)(t) where u_(diff1)(t) represents avoltage obtained by summing energy in the arms of the leg andu_(diff2)(t) represents a voltage obtained by calculating a differencein energy between the arms of the leg.
 9. An apparatus for calculatinginsertion indices for a phase leg of a DC to AC modular multilevelconverter, the converter comprising one phase leg between upper andlower DC source common bars for each phase, each phase leg comprisingtwo serially connected arms, wherein an AC output for each phase leg isconnected between the two serially connected arms, wherein each armcomprises a number of submodules, wherein each submodule can be in abypass state or a voltage insert mode, the insertion index comprisingdata representing a portion of available submodules that should be inthe voltage insert mode for a particular arm, the apparatus comprises: acontroller arranged to calculate a desired arm voltage for an upper armconnected to the upper DC source common bar and a lower arm connected tothe lower DC source common bar, to obtain values representing actualtotal arm voltages in the upper arm and lower arm, respectively, and tocalculate insertion indices for the upper and lower arm, respectively,using the respective desired arm voltage and the respective valuerepresenting the total actual arm voltage; wherein the calculating ofdesired arm voltages for a phase leg comprises calculatingu _(CU)(t)=u _(D)/2−e _(V)(t)−u _(diff)(t) for the upper arm, andcalculatingu _(CL)(t)=u _(D)/2+e _(V)(t)−u _(diff)(t) for the lower arm, whereu_(CU)(t) represents desired upper arm voltage, u_(CL)(t) representsdesired lower arm voltage, u_(D) represents a voltage between the upperand lower DC source common bars, e_(V)(t) represents a reference innerAC output voltage and u_(diff)(t) represents a control voltage tocontrol a current, iV(t), passing through the whole phase leg, whereinthe step of obtaining values representing actual arm voltages comprises:calculating u_(CU) ^(Σ)(t), actual total voltage for the upper arm,using C_(arm), capacitance for the upper arm, î_(diff0), DC currentpassing through the two serially connected arms of the phase leg, W_(CU)^(Σ)(t), desired average energy in the upper arm, ê_(V), amplitude ofreference for the inner AC output voltage, î_(V), amplitude of AC outputcurrent, φ, a phase difference between i_(V)(t) and e_(V)(t), andcalculating u_(CL) ^(Σ)(t), actual total voltage for the lower arm,using C_(arm), capacitance for the lower arm, î_(diff0), DC currentpassing through the two serially connected arms of the phase leg, W_(CL)^(Σ)(t), desired average energy in the lower arm, ê_(V), amplitude ofreference for inner AC output voltage, î_(V), amplitude of AC outputcurrent φ, a phase difference between i_(V)(t) and e_(V)(t).
 10. Theapparatus of claim 9, wherein the step of obtaining a value representingactual arm voltage comprises calculating${\hat{i}}_{{diff}\; 0} = \frac{{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}{u_{D} + \sqrt{u_{D}^{2} - {4\mspace{11mu} R{\hat{e}}_{v}{\hat{i}}_{v}\cos\;\varphi}}}$where R represents the resistance of the converter arm.
 11. Theapparatus of claim 10, wherein the step of obtaining a valuerepresenting actual arm voltage comprises calculating${u_{CU}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CU}^{\Sigma}(t)}}{C_{arm}}}$where W_(CU) ^(Σ)(t) is calculated as follows:${W_{CU}^{\Sigma}(t)} = {W_{{CU}\; 0}^{\Sigma} - {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} + {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$and calculating${u_{CL}^{\Sigma}(t)} = \sqrt{\frac{2{W_{CL}^{\Sigma}(t)}}{C_{arm}}}$where W_(CL) ^(Σ)(t) is calculated as follows:${W_{CL}^{\Sigma}(t)} = {W_{{CL}\; 0}^{\Sigma} + {\frac{{\hat{e}}_{v}{\hat{i}}_{{diff}\; 0}}{\omega_{1}}\sin\;\omega_{1}t} - {\left( {\frac{u_{D}}{2} - {R\;{\hat{i}}_{{diff}\; 0}}} \right)\frac{{\hat{i}}_{V}}{2\omega_{1}}{\sin\left( {{\omega_{1}t} + \varphi} \right)}} - {\frac{{\hat{e}}_{V}{\hat{i}}_{V}}{8\omega_{1}}{\sin\left( {{2\omega_{1}t} + \varphi} \right)}}}$where ω₁ represents the angular velocity of the network frequency.